In calculus courses like MATH 140 and 141, students learn to calculate the derivative of a function and to use derivatives in simple applications. In MATH 251, students will learn how derivatives commonly appear in equations used to describe the world. Equations involving derivatives are called differential equations. Differential equations play an important role in modeling the real world. Newton's laws, Maxwell's laws of electromagnetism, Einstein's equations of general relativity, Euler and Bernoulli's beam equation, the Black-Scholes equation from finance, Perelson's viral-dynamics equations in biology, and the million-dollar Navier-Stokes equations are all differential equations used daily in their respective disciplines. Today, differential equations are one of the fundamental mathematical tools for the study of systems that change over time, and are used in most areas of science, engineering, and mathematics.

MATH 251 is an introductory course on ordinary differential equations and partial differential equations. It is a 1-credit extension of MATH 250; this extra credit is used to allow the coverage of partial differential equations (which are not covered in MATH 250 because of time limitations). Partial differential equations are differential equations that involve derivatives with respect to more than one independent variable. Such equations are needed to understand phenomena like the vibration of a guitar string, the failure of an I-beam, and the diffusion of particles in fluid. The goal of this course is to teach the students some elementary techniques of ordinary and partial differential equations. Some of the topics covered in this course include first order ordinary differential equations, second order ordinary differential equations with constant coefficients, 2 × 2 linear systems with constant coefficients, stability of equilibrium solutions, Laplace transforms, Fourier series, partial differential equations which include the heat equation, wave equation and the Laplace equation.

MATH 251 fulfills all the differential equations training requirements fulfilled by MATH 250 and some majors require 251 in place of 250. On completing MATH 251, students may enroll in MATH 405, 411, 412, 417, and 419.

Class size, frequency of offering, and evaluation methods will vary by location and instructor. For these details check the specific course syllabus.

An honors version of this course is offered at least once per year.

- FAll 2015
- Spring 2014 (251.101)

**Exams**